-00)0) 2 (AB 22) Let L : R, R2 be a linear transformation. You are given that L 2- 3 (a) Find the matrix A that represents L with respect to the basisu-| | 2-1 1-1 4 1 and the 6 standard basis F1 (b) Find the matrix B that represents IL with respect to the standard basis in both R3 and R2
3. This example hopes to illustrate why the vector spaces the linear transformation are defined on are critical to the question of invertibility. Let L : → p, be defined by L(p)(t+1)p(t)-plt). (a) Given a basis of your choice, find a matrix representation of I with respect to your chosen basis (b) Show L: P+P is not invertible (e) Let V-span+21-4,+2t-8). It can be shown that L VV. Given an ordered basis for V of your choice, find a matrix...
Let L: R3 --> R3 be defined by
Only need c-e solved.
6, (24 points) Let L : R3 → R3 be defined by (a) Find A, the standard matrix representation of f (b) Let 0 -2 2. Check that倔,G, u) is a basis of R3. (c) Find the transition matrix B from the ordered basis U (t, iz, a) to the standard basis {e, е,6). For questions (d) and (e), you can write your answer in terms of A...
[22 Q4. Let L: R — be a transformation defined by L - -33 (a) Show that L is a linear transformation. (8 pts) (1) Find the standard matrix A of L, and find 2 (1 : 1) using the matrix A. (7 pts) (e) Do you think that any transformation TR is linear? (Justify your answer) (5 pts)
Problem 13. Let l be the line in R' spanned by the vector u = 3 and let P:R -R be the projection onto line l. We have seen that projection onto a line is a linear transformation (also see page 218 example 3.59). a). Find the standard matrix representation of P by finding the images of the standard basis vectors e, e, and e, under the transformation P. b). Find the standard matrix representation of P by the second...
Q4. Let L: R2 + Rº be a transformation defined by L (0-2 [3u2 – U1 U1 – U2 -502 (a) Show that I is a linear transformation. (b) Find the standard matrix A of L, and find L ([31]) using the matrix A. (c) Do you think that any transformation T:R2 + R² is linear? (Justify your answer).
Let x = [xı x2 x3], and let TER → R be the linear transformation defined by T() = x1 + 6x2 – x3 -X2 X1 + 4x3 Let B be the standard basis for R2 and let B' = {V1, V2, V3}, where 7 7 and v3 = 7 V1 V2 [] --[] 0 Find the matrix of I with respect to the basis B. and then use Theorem 8.5.2 to compute the matrix of T with respect to...
Problem 3 Let L: R4 → R3 be given by L (6)-1 (3:01 - 4.12 + 1104) (15.12 + 9.23 - 21:04) 6.01 +9.12 + 4.13 - 5.14) a) (4 pts] Show that L is a linear transformation, and find the matrix representation A of L with respect to the standard bases for R' and R3. b) [3 pts] Use part a) to find a basis for ker(L). c) [3 pts] Use part a) for find a basis for im(L).
1. Let L: P1(R) + P1(R) be a linear transformation given by L(a + bx) = a - b + (2a – b)x. Let S = {1, 2} and T = {1+x} be two basis for P1(R). (a) Find the matrix A of L with respect to basis S. (a) Find the matrix B of L with respect to basis T. (c) Find the matrix P obtained by expressing vectors in basis T in terms of vectors in basis (d)...
6. Let L be the linear operator mapping R3 into R3 defined by L(x) Ax, where A=12 0-2 and let 0 0 Find the transition matrix V corresponding to a change of basis from i,V2. vs) to e,e,es(standard basis for R3), and use it to determine the matrix B representing L with respect to (vi, V2. V